Calculus Examples

Solve the Differential Equation ydx=2(x+y)dy
Step 1
Rewrite the differential equation to fit the Exact differential equation technique.
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Step 1.1
Subtract from both sides of the equation.
Step 2
Find where .
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Step 2.1
Differentiate with respect to .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 3
Find where .
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Step 3.1
Differentiate with respect to .
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
By the Sum Rule, the derivative of with respect to is .
Step 3.4
Differentiate using the Power Rule which states that is where .
Step 3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.6
Simplify the expression.
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Step 3.6.1
Add and .
Step 3.6.2
Multiply by .
Step 4
Check that .
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Step 4.1
Substitute for and for .
Step 4.2
Since the left side does not equal the right side, the equation is not an identity.
is not an identity.
is not an identity.
Step 5
Find the integration factor .
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Step 5.1
Substitute for .
Step 5.2
Substitute for .
Step 5.3
Substitute for .
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Step 5.3.1
Substitute for .
Step 5.3.2
Subtract from .
Step 5.3.3
Substitute for .
Step 5.4
Find the integration factor .
Step 6
Evaluate the integral .
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Step 6.1
Since is constant with respect to , move out of the integral.
Step 6.2
Since is constant with respect to , move out of the integral.
Step 6.3
Multiply by .
Step 6.4
The integral of with respect to is .
Step 6.5
Simplify.
Step 6.6
Simplify each term.
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Step 6.6.1
Simplify by moving inside the logarithm.
Step 6.6.2
Exponentiation and log are inverse functions.
Step 6.6.3
Rewrite the expression using the negative exponent rule .
Step 7
Multiply both sides of by the integration factor .
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Step 7.1
Multiply by .
Step 7.2
Cancel the common factor of .
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Step 7.2.1
Factor out of .
Step 7.2.2
Cancel the common factor.
Step 7.2.3
Rewrite the expression.
Step 7.3
Multiply by .
Step 7.4
Apply the distributive property.
Step 7.5
Multiply by .
Step 7.6
Factor out of .
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Step 7.6.1
Factor out of .
Step 7.6.2
Factor out of .
Step 7.6.3
Factor out of .
Step 7.7
Factor out of .
Step 7.8
Factor out of .
Step 7.9
Factor out of .
Step 7.10
Rewrite as .
Step 7.11
Move the negative in front of the fraction.
Step 8
Set equal to the integral of .
Step 9
Integrate to find .
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Step 9.1
Apply the constant rule.
Step 9.2
Combine and .
Step 10
Since the integral of will contain an integration constant, we can replace with .
Step 11
Set .
Step 12
Find .
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Step 12.1
Differentiate with respect to .
Step 12.2
By the Sum Rule, the derivative of with respect to is .
Step 12.3
Evaluate .
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Step 12.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 12.3.2
Rewrite as .
Step 12.3.3
Differentiate using the chain rule, which states that is where and .
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Step 12.3.3.1
To apply the Chain Rule, set as .
Step 12.3.3.2
Differentiate using the Power Rule which states that is where .
Step 12.3.3.3
Replace all occurrences of with .
Step 12.3.4
Differentiate using the Power Rule which states that is where .
Step 12.3.5
Multiply the exponents in .
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Step 12.3.5.1
Apply the power rule and multiply exponents, .
Step 12.3.5.2
Multiply by .
Step 12.3.6
Multiply by .
Step 12.3.7
Raise to the power of .
Step 12.3.8
Use the power rule to combine exponents.
Step 12.3.9
Subtract from .
Step 12.4
Differentiate using the function rule which states that the derivative of is .
Step 12.5
Simplify.
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Step 12.5.1
Rewrite the expression using the negative exponent rule .
Step 12.5.2
Combine terms.
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Step 12.5.2.1
Combine and .
Step 12.5.2.2
Move the negative in front of the fraction.
Step 12.5.2.3
Combine and .
Step 12.5.2.4
Move to the left of .
Step 12.5.3
Reorder terms.
Step 13
Solve for .
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Step 13.1
Solve for .
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Step 13.1.1
Move all terms containing variables to the left side of the equation.
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Step 13.1.1.1
Add to both sides of the equation.
Step 13.1.1.2
Combine the numerators over the common denominator.
Step 13.1.1.3
Apply the distributive property.
Step 13.1.1.4
Add and .
Step 13.1.1.5
Add and .
Step 13.1.1.6
Cancel the common factor of and .
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Step 13.1.1.6.1
Factor out of .
Step 13.1.1.6.2
Cancel the common factors.
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Step 13.1.1.6.2.1
Factor out of .
Step 13.1.1.6.2.2
Cancel the common factor.
Step 13.1.1.6.2.3
Rewrite the expression.
Step 13.1.2
Subtract from both sides of the equation.
Step 14
Find the antiderivative of to find .
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Step 14.1
Integrate both sides of .
Step 14.2
Evaluate .
Step 14.3
Since is constant with respect to , move out of the integral.
Step 14.4
Since is constant with respect to , move out of the integral.
Step 14.5
Multiply by .
Step 14.6
Move out of the denominator by raising it to the power.
Step 14.7
Multiply the exponents in .
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Step 14.7.1
Apply the power rule and multiply exponents, .
Step 14.7.2
Multiply by .
Step 14.8
By the Power Rule, the integral of with respect to is .
Step 14.9
Simplify the answer.
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Step 14.9.1
Rewrite as .
Step 14.9.2
Simplify.
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Step 14.9.2.1
Multiply by .
Step 14.9.2.2
Combine and .
Step 15
Substitute for in .